Show that the coefficient of the middle term in the expansion of `(1+x)^(2n)`is equal to the sum of the coefficients of two middle terms in the expansion of `(1+x)^(2n-1)`.

Options

A. A+B=C
B. B+C=A
C. C+A= B
D. A+B +C = 0

Answer

b

Explanation

Clearly, `(n+1)^(th)` term is the middle term in the
expansion of `(1 + x)^(2n)` and its coefficients is `""^(2n)C_(n)` i.e. A = `""^(2n)C_(n)`
In the expansion of `(1 +x)^(2n-1), n^(th) and (n +1)^(th)` terms are
middle terms and their coefficient are `""^(2n)C_(n-1) and ""^(2n-1)C_(n)`
respectively i.e. B `""^(2n-1)C_(n-1) and C = ""^(2n)C_(n)`,
We have,
`""^(2n-1)C_(n-1)+""^(2n-1)C_(n) = ""^((2n-1)+1)C_(n)[because ""^(n)C_(r-1) +""^(n)C_(r) = ""^(n+1)C_(r)]`
`therefore B + C = A`